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On the geometry of frame bundles

Kamil Niedziałomski (2012)

Archivum Mathematicum

Let ( M , g ) be a Riemannian manifold, L ( M ) its frame bundle. We construct new examples of Riemannian metrics, which are obtained from Riemannian metrics on the tangent bundle T M . We compute the Levi–Civita connection and curvatures of these metrics.

On the multiplicity of eigenvalues of conformally covariant operators

Yaiza Canzani (2014)

Annales de l’institut Fourier

Let ( M , g ) be a compact Riemannian manifold and P g an elliptic, formally self-adjoint, conformally covariant operator of order m acting on smooth sections of a bundle over M . We prove that if P g has no rigid eigenspaces (see Definition 2.2), the set of functions f C ( M , ) for which P e f g has only simple non-zero eigenvalues is a residual set in C ( M , ) . As a consequence we prove that if P g has no rigid eigenspaces for a dense set of metrics, then all non-zero eigenvalues are simple for a residual set of metrics in the C -topology....

Prescribing a fourth order conformal invariant on the standard sphere, part II : blow up analysis and applications

Zindine Djadli, Andrea Malchiodi, Mohameden Ould Ahmedou (2002)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

In this paper we perform a fine blow up analysis for a fourth order elliptic equation involving critical Sobolev exponent, related to the prescription of some conformal invariant on the standard sphere ( 𝕊 n , h ) . We derive from this analysis some a priori estimates in dimension 5 and 6 . On 𝕊 5 these a priori estimates, combined with the perturbation result in the first part of the present work, allow us to obtain some existence result using a continuity method. On 𝕊 6 we prove the existence of at least one...

Selfdual spaces with complex structures, Einstein-Weyl geometry and geodesics

David M J. Calderbank, Henrik Pedersen (2000)

Annales de l'institut Fourier

We study the Jones and Tod correspondence between selfdual conformal 4 -manifolds with a conformal vector field and abelian monopoles on Einstein-Weyl 3 -manifolds, and prove that invariant complex structures correspond to shear-free geodesic congruences. Such congruences exist in abundance and so provide a tool for constructing interesting selfdual geometries with symmetry, unifying the theories of scalar-flat Kähler metrics and hypercomplex structures with symmetry. We also show that in the presence...

Smooth metric measure spaces, quasi-Einstein metrics, and tractors

Jeffrey Case (2012)

Open Mathematics

We introduce the tractor formalism from conformal geometry to the study of smooth metric measure spaces. In particular, this gives rise to a correspondence between quasi-Einstein metrics and parallel sections of certain tractor bundles. We use this formulation to give a sharp upper bound on the dimension of the vector space of quasi-Einstein metrics, providing a different perspective on some recent results of He, Petersen and Wylie.

Spezielle äquiforme Zwangläufe

Helmut Pottmann (1984)

Aplikace matematiky

In der Ebene kann ein äquiformer Zwanglauf so bestimmt werden, daß jede Gerade einer beweglichen Ebene in einer festen Ebene eine zykloidale Kurve mit demselben Modul umhüllt. Das Problem wird ebenfalls im Raum gelöst und verallgemeinert.

Structure of the kernel of higher spin Dirac operators

Martin Plechšmíd (2001)

Commentationes Mathematicae Universitatis Carolinae

Polynomials on n with values in an irreducible Spin n -module form a natural representation space for the group Spin n . These representations are completely reducible. In the paper, we give a complete description of their decompositions into irreducible components for polynomials with values in a certain range of irreducible modules. The results are used to describe the structure of kernels of conformally invariant elliptic first order systems acting on maps on n with values in these modules.

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